Kolmogorov Complexity in perspective. Part I: Information Theory and Randomnes

TL;DR

This paper surveys various approaches to information, focusing on Kolmogorov complexity's role in understanding randomness and classification within information theory, highlighting historical developments and recent advances.

Contribution

It provides a comprehensive overview of Kolmogorov complexity's application to randomness and classification, emphasizing its theoretical foundations and historical context.

Findings

Kolmogorov complexity formalizes randomness in information theory.

Historical development from 1960s to present is outlined.

Applications include randomness testing and data classification.

Abstract

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts in the same volume. Part I is dedicated to information theory and the mathematical formalization of randomness based on Kolmogorov complexity. This last application goes back to the 60's and 70's with the work of Martin-L\"of, Schnorr, Chaitin, Levin, and has gained new impetus in the last years.